Pressure-induced reversal of Peierls-like distortions elicits the polyamorphic transition in GeTe and GeSe

While polymorphism is prevalent in crystalline solids, polyamorphism draws increasing interest in various types of amorphous solids. Recent studies suggested that supercooling of liquid phase-change materials (PCMs) induces Peierls-like distortions in their local structures, underlying their liquid-liquid transitions before vitrification. However, the mechanism of how the vitrified phases undergo a possible polyamorphic transition remains elusive. Here, using high-energy synchrotron X-rays, we can access the precise pair distribution functions under high pressure and provide clear evidence that pressure can reverse the Peierls-like distortions, eliciting a polyamorphic transition in GeTe and GeSe. Combined with simulations based on machine-learned-neural-network potential, our structural analysis reveals a high-pressure state characterized by diminished Peierls-like distortion, greater coherence length, reduced compressibility, and a narrowing bandgap. Our finding underscores the crucial role of Peierls-like distortions in amorphous octahedral systems including PCMs. These distortions can be controlled through pressure and composition, offering potentials for designing properties in PCM-based devices.


Overview of data processing
The flowchart of the data processing is shown in Supplementary Fig. 1.First, we classified the measured diffraction data I(Q) into two groups based on the existence of Bragg peaks.The diffraction data without Bragg peaks were directly transferred for further data processing.For the partially crystallized data with Bragg peaks, we performed Rietveld refinement to split the I(Q) into the broad scattering predominantly from the amorphous state and the Bragg peaks from the crystalline state.Next, we applied the background subtraction, intensity correction, and normalization for the I(Q) of the amorphous state to convert it into the total structure factor S(Q). Finally, the reduced pair distribution function G(r) was obtained by the Fourier transformation of the S(Q).

Rietveld refinement
For the partially crystallized data of high-pressure GeTe, Rietveld refinement was performed to decompose the I(Q) into the crystalline contribution and the remaining broad scattering predominantly from the amorphous state.The data analysis was performed on the program Synchrotron Powder and Plotpro 1 .The analytical range was selected as 0.4° to 20° in 2.The split-type pseudo-Voigt function was used as the profile function for modelling the Bragg peaks.The remaining broad scattering was expressed as the sum of 9 profile functions.We employed the split-type PearsonVII functions for describing the asymmetric shape of the broad amorphous peaks.The profile parameters were optimized during the Rietveld refinement to achieve the best fit.Supplementary Fig. 2 shows the representative results of Rietveld refinement.As shown in Supplementary Fig. 2a, Bragg peaks at high-pressure conditions are modelled by the rock-salt type cubic phase (space group 3 �  ) of GeTe.The difference between the observed and calculated intensity of Bragg peaks was only 0.2 % on average for the first 16 peaks (located at < 8°), which constitutes roughly 80 % of the total crystalline contribution.A slight deviation around 3.3° can be attributed to the scattering from a component of the sample cell.The agreement of overall analytical range was provided by the reliability factor Rwp = 4.18 % (lower values indicates the better reliability), suggesting that the remaining broad contribution was successfully expressed as the sum of 9 profile functions.Based on these results, we could successfully decompose the I(Q) into the crystalline contribution and the remaining broad scattering with sufficient accuracy.We note that the cubic phase was transformed into the stable rhombohedral phase (space group 3 � ) at ambient pressure after pressure release, as clearly demonstrated by the peak split around 3.6° and the smaller separation between the 1 st and 2 nd peaks with respect to the cubic phase (Supplementary Fig. 2b) 2 .

Total structure factor S(Q) and reduced pair distribution function G(r)
Proper corrections and normalization were applied to the I(Q) (raw data or the extracted broad contribution from Rietveld refinement) to obtain the total structure factor S(Q) and reduced pair distribution function G(r).All the procedures were performed on the software pdfgetX2 3 .The contribution from sample environment was subtracted from the scattering data.The profile of the sample environment was measured from an empty CDT cell (Supplementary Fig. 3a).In the present experimental setup, collimation slits placed in front of the sample can block the scattering from the sample environment such as the cell and surrounding air, achieving considerably low background intensity 4 .As shown in Supplementary Fig. 3a, the I(Q) of the empty cell suggested a small contribution from the sample environment, which was mainly composed of air scattering (less than 3 % above Q > 2.0 Å -1 with respect to the intensity of sample scattering).The corrections for self-absorption, multiple scattering, and oblique incidence were applied to the I(Q) to convert it into S(Q).The energy-dependent Compton scattering (1/E quadratic form) was calculated.Supplementary Fig. 3a shows the I(Q) after correction and the profile of Compton scattering.A Breit-Dirac factor of 2 was used for the Compton scattering.The parameters of Compton scattering were optimized to achieve S(Q)→1 for the Q-range from 20 to 25 Å -1 .Supplementary Fig. 3b shows the obtained total structure factor S(Q), which was subsequently Fourier-transformed into the reduced pair distribution function G(r).The Lorch function was used for reducing the truncation oscillations in G(r).The peak positions of S(Q) and G(r), considered in the present study, are known to be robust with respect to the selection of the correction parameters 5 .

The full width of half maximum (FWHM) of the first main diffraction peak of S(Q)
The evaluation of FWHM of the first main diffraction peak Q1 in S(Q) is interfered with the partial overlap with the neighboring second peak, which shifts towards the first peak with pressure increase.In the main text, FWHM is evaluated as the distance between two positions of Q at which S(Q) becomes a half of S(Q1), possibly including some contribution from the second peak.To examine the robustness of the pressure dependence against the peak overlap, FWHM is estimated by fitting the first peak with a Lorentzian profile 6 .Supplementary Fig. 5a shows the representative results of fitting with two different Q-ranges: 1.5 Å -1 to Q1 (denoted as low-Q) and 1.5 Å -1 to 2.5 Å -1 (high-Q).The fitting with low-Q range provided the lowest FWHM of 0.478 Å -1 , whereas half of S(Q1) resulted in the largest FWHM of 0.632 Å -1 .Supplementary Fig. 5b and 5c shows the pressure dependence of FWHM for GeSe and GeTe estimated by these three methods.Clearly, regardless of such a difference in absolute values of FWHM, the width exhibited essentially the same pressure dependence., i.e. a kink is observed around the transition pressure 3.5 GPa for GeSe, and FWHM of GeTe exhibited a distinct drop above 1.8 GPa.The results demonstrate that the coherent length is significantly increased after polyamorphic transition and the high-pressure state of GeSe and GeTe is characterized by the higher structure coherence.

The pressure environment of the Paris-Edinburgh press
The high-pressure X-ray scattering data in the present study were collected from the samples in the Paris-Edinburgh press.We used soft hexagonal BN and MgO as a pressure medium surrounding the sample, which is often used in large volume press experiments for quasi-hydrostatic environment.To investigate the effect of homogeneity of pressure environment, we compared the pressure dependence of the atomic volume ratio V/V0 of the partially crystallized volume fraction of GeTe obtained in this study with the previous ones which were obtained under hydrostatic pressure conditions (Supplementary Fig. 6).V is the atomic volume calculated from the lattice parameters of cubic GeTe, and V0 is the atomic volume at ambient pressure.The solid line is the third-order Birch-Murnaghan fit reported in the previous study under the hydrostatic pressure conditions by using the diamond anvil cell and ethanol-methanol-water mixture as a pressure medium 2 .In the present study, we observed the crystalline peaks of GeTe only above 3.4 GPa in the compression process.To compare the pressure response, the V0 was estimated by using the volume ratio of the reported value at 3.4 GPa.All the data points are consistent up to 10.2 GPa with the reported curve within error bar, suggesting that the observed pressure response in the present pressure range should be close to the one that could have been observed under hydrostatic conditions.